8 Then God spoke to Noah and to his sons with him, saying:
9 “And as for Me, behold, I establish My covenant with you and with your descendants after you,
10 and with every living creature that is with you: the birds, the cattle, and every beast of the earth with you, of all that go out of the ark, every beast of the earth.
11 Thus I establish My covenant with you: Never again shall all flesh be cut off by the waters of the flood; never again shall there be a flood to destroy the earth.”
12 And God said: “This is the sign of the covenant which I make between Me and you, and every living creature that is with you, for perpetual generations:
13 I set My rainbow in the cloud, and it shall be for the sign of the covenant between Me and the earth.
14 It shall be, when I bring a cloud over the earth, that the rainbow shall be seen in the cloud;
15 and I will remember My covenant which is between Me and you and every living creature of all flesh; the waters shall never again become a flood to destroy all flesh.
16 The rainbow shall be in the cloud, and I will look on it to remember the everlasting covenant between God and every living creature of all flesh that is on the earth.”
17 And God said to Noah, “This is the sign of the covenant which I have established between Me and all flesh that is on the earth.”
Man’s obedience to these commands (Genesis 9:1-7) was not a condition determining whether God would keep His part of the bargain. God promised unconditionally … that He would never again send a worldwide flood, or destroy all flesh, as long as the earth remained. — Morris, page 227
It is significant that the Noahic covenant was not only with Noah and his descendants, but also with the animals going out of the Ark and their descendants. Even though animals do not possess an eternal soul and spirit, as men do, they are God’s creatures; and He is concerned about them. — Morris, page 228
It is not only that man himself would see the rainbow. God also would “look upon it,” whenever He would “bring a cloud over the earth, ” and would “remember His covenant.” This was peculiarly “my bow,” according to the Lord, probably referring to the fact that it had just now been formed as a result of the great Flood which He had brought on the earth.
… His covenant was an everlasting covenant. It was valid for [those who had been on the Ark] and for their children to “perpetual generations,” until God’s great promised time of consummation and restoration of all things.
The rainbow appears only three more times in Scripture. Once, in Ezekiel 1:28, the rainbow is seen surrounding the throne of God as He prepares to visit judgment on His people Israel. Again, the rainbow is seen around His throne just before the coming Great Tribulation in Revelation 4:3. In both these cases, the picture is one of imminent judgment and suffering, bot only limited judgment and suffering, with God’s grace ruling over all.
Finally, when the mighty angel of Revelation 10:1, who can none other than the Lord Jesus Christ Himself, comes to claim dominion over the world, which He had created but which had long been under the dominion of the wicked one. He is accompanied by the same “seven thunders” of judgment which apparently had once cried forth at the time of the Flood (Revelation 10:3-4, compared with Psalm 29:3-10). And instead of a crown of thorns, which once He wore as He bore the Curse for us, the Word says there will be “the rainbow upon His head.” The definite article is in the original: the rainbow. This can hardly refer to any other rainbow than to “My bow,” the token of the everlasting covenant between God and all flesh (Genesis 10:16, with Revelation 10:6). — Morris, pages 229-231
It is happy to bear in mind, that when the bow appears, the eye of God rests upon it; and man is cast not upon his own imperfect and most uncertain memory, but upon God’s. “I,” says God, “will remember.” How sweet to think of what God will, and what He will not, remember! He will remember His covenant, but He will not remember His people’s sins. The cross, which ratifies the former, puts away the latter. — Mackintosh, page 110